The quasiparticle band structure and optical properties of single-walled zigzag and armchair SiC nanotubes (SiC-NTs) as well as a single SiC sheet are investigated by ab initio many-body calculations using the GW and the GW plus Bethe-Salpeter equation approaches, respectively. Significant GW quasiparticle corrections, of more than 1.0 eV, to the Kohn-Sham band gaps from the local density approximation (LDA) calculations are found. The GW self-energy corrections transform the SiC sheet from an indirect LDA band gap to a direct band gap material. Furthermore, the quasiparticle band gaps of SiC-NTs with different chiralities behave very differently as a function of tube diameter, and this can be attributed to the difference in the curvature-induced orbital rehybridization among the different chiral nanotubes. The calculated optical absorption spectra are dominated by discrete exciton peaks due to exciton states with a high binding energy, up to 2.0 eV, in the SiC sheet and SiC-NTs. The formation of strongly bound excitons is attributed to the enhanced electron-hole interaction in these low-dimensional systems. Remarkably, the excited electron amplitude of the exciton wave function is found to peak on Si atoms near the hole position (which is on the C site) in zigzag SiC-NTs, indicating a charge transfer from an anion (hole) to its neighboring cations by photoexcitation. In contrast, this pronounced peak structure disappears in the exciton wave function in armchair SiC-NTs. Furthermore, in armchair SiC-NTs, the bound exciton wave functions are more localized and also strongly cylindrically asymmetric. The high excitation energy, ∼3.0 eV, of the first bright exciton, with no dark exciton below it, suggests that small-radius armchair SiC-NTs could be useful for optical devices working in the UV regime. On the other hand, zigzag SiC-NTs have many dark excitons below the first bright exciton and hence may have potential applications in tunable optoelectric devices ranging from infrared to UV frequencies by external perturbations.